One Rule, Two Graphs

Follow the same value

There are two marked points here, but only one angle and one radius. Begin with angle zero and radius four. The polar point lies four units to the right of the origin. In the small graph, angle zero is at the left edge and radius four is the vertical value. The two pictures organize the same information differently.

Watch both markers as the angle advances to a quarter turn. In the small graph, the marker travels to the right and downward: angle increases while radius decreases to two. In the polar picture, the direction turns upward. The point ends above the origin, even though the radius graph has been descending.

A descending radius graph therefore does not mean the polar point must move downward. Its vertical axis measures signed radius, not the polar point’s height. At this quarter turn, radius two puts the polar point two units above the origin. The equation supplies a distance instruction, which the selected direction then interprets.

Continue toward half a turn. The radius graph descends to zero while the polar point approaches the origin. These events do correspond directly: zero radius always places the polar point at the origin. The graph’s horizontal coordinate tells us which angle produced that zero, even when the polar location itself is the same.

Read the radius graph

Let us enlarge the rectangular graph. Read across it as a sequence of angle values, with radius measured vertically. We are still using r equals two plus two cosine theta. The maximum is four at the beginning and end of the turn; the minimum is zero at half a turn. The horizontal gold line marks the constant contribution, two.

Follow the marker from angle zero to half a turn. Its height falls from four through two to zero. The oscillating contribution starts two above the gold line and ends two below it. That vertical distance from the midline is the contribution of two cosine theta, which can be positive, zero or negative.

Now follow the second half. The marker climbs from zero back to four. At three quarters of a turn its radius is two again. Equal heights in this graph mean equal signed radii. They do not mean equal locations in the polar picture, because the angles can differ.

Return to both views and stop at three quarters of a turn. The marked polar point is below the origin, at radius two. Earlier, the same radius placed it above the origin at a quarter turn. The rectangular graph keeps those two angle inputs separated horizontally, even though their output heights agree.

Bend one representation into the other

Keep the angle at a quarter turn and watch the large picture unwrap. The equation and its current radius stay fixed. The moving construction changes how the angle positions are arranged in space. Follow the bright marker as the curved representation opens into a strip; its height ends at radius two.

In this large flattened strip, angle increases toward the left, as its label indicates. The small graph increases toward the right. That reversal of horizontal orientation is part of the wrapping construction; it does not change the equation. In both views, the vertical value is still radius, and the gold baseline is two.

Move the angle along the strip toward half a turn. The bright marker reaches height zero, just as it does in the small graph. We are seeing the same zero through two layouts. The flattened strip has room to display different angle positions separately before we bend them back around the origin.

Wrap the strip back into the polar curve. The zero-radius marker arrives at the origin. The intermediate bent shapes are a visual connection between representations, not extra definitions of polar coordinates. At the polar endpoint, the familiar rule applies again: choose direction theta and place the point at signed radius r.

Below zero means reverse

Change the rule to one plus two cosine theta and begin at angle zero. Its radius starts at three, and its minimum will be minus one. The small graph now crosses its zero line. The polar outline has an inner loop. We can connect that loop with the interval where the graph is below zero.

Follow both markers to one hundred twenty degrees. The radius reaches zero for the first time. In the small graph the marker crosses the horizontal zero line; in the polar picture it reaches the origin. Watch the sign rather than guessing from the shape of the full outline.

Continue to half a turn. The rectangular marker reaches radius minus one, below zero. The polar marker sits to the right of the origin because a negative radius reverses the leftward direction. Below zero in one picture can mean right of the origin in the other. The axes are answering different questions.

Continue to the second zero at two hundred forty degrees. The polar marker returns to the origin after traveling around the inner loop. Its companion in the small graph has crossed the entire below-zero interval. That interval explains the loop through signed distance, not through a resemblance between the two drawn shapes.

Offset and amplitude do different jobs

Compare the rule three plus two cosine theta. The graph’s midline is now three and its range runs from one to five. Its radius never reaches zero, so the polar curve avoids the origin. Moving the constant upward changed every output radius by the same amount; it did not change the oscillation’s amplitude.

Now use two plus one cosine theta. The midline is two and the amplitude is one, giving radii from one to three. Compared with the original cardioid, the minimum has risen while the maximum has fallen. Changing amplitude draws the graph toward its midline; it is different from shifting that entire graph upward.

Trace this smaller-amplitude rule. The radius remains positive through the full turn, moving between three and one. The marker never reaches the origin. You can predict that absence of an origin visit from the rectangular graph before tracing every angle of the polar curve.

Restore two plus two cosine theta and trace it again. Its minimum returns to zero, and the cusp returns to the origin. The coefficients now have visible roles: the constant sets the rectangular midline, and the cosine coefficient sets the vertical oscillation around it. The polar shape follows from those radius values.

Hold the angle while changing the rule

Keep the angle at sixty degrees and read the two markers together. The rule is two plus two cosine theta. Cosine of sixty degrees is one half, so the radius is three. The point lies one and a half units to the right and about two point six units above the origin. The small graph instead locates sixty degrees horizontally and three vertically.

Hold that angle and increase the constant from two to three. Every height of the radius graph rises by one. Our selected radius rises from three to four, while the selected angle stays fixed. In the polar picture, the point moves outward along the same direction. The whole polar curve changes, but it is not a simple upward translation of the old curve.

Enlarge the radius graph and keep its constant at three. Now decrease the cosine coefficient from two to one. At our sixty-degree input, the oscillating contribution falls from one to one half, so the radius falls from four to three and a half. The midline stays at three. This change squeezes the oscillation around its midline instead of lifting every height equally.

Return to both pictures and move to a quarter turn. The cosine contribution is now zero, so the radius equals the constant, three. The polar point stands three units above the origin. This angle gives us a useful test: changing the cosine coefficient should change the overall curve while leaving this particular point in the same place.

Increase that coefficient from one to three while keeping the quarter-turn angle. Watch the curve spread and its minimum approach zero. The selected point remains three units above the origin, and its radius-graph marker remains at height three. A control can change the complete graph without changing the output at every input. Here the zero cosine factor explains the stationary selected point.

At this same angle, lower the constant from three to one. Now the selected radius does change, falling from three to one, and the polar point moves down toward the origin along its fixed ray. Compare the two experiments: the cosine coefficient had no effect at this input, but the constant changed its output directly. Read the equation and the selected values together before predicting a movement.

More oscillations, different paths

For a final comparison, set the constant to zero and use r equals two cosine of two theta. The small graph now contains two full oscillations during one complete turn of direction. The inner two multiplies the angle inside cosine; it changes how quickly the radius oscillates, not the physical speed at which you must move a control.

Follow the first quarter turn. The radius changes from plus two through zero to minus two. At ninety degrees the direction points up, but the negative radius places the point two units down. Compare the low point of the rectangular graph with that lower polar endpoint.

Complete the circuit and watch the four petals emerge in the bright trace. Two oscillations in the radius graph have produced four petals here, because positive and negative radii place points in different directions. The next lesson will examine that relationship carefully, including why odd frequencies behave differently.

Return to the cardioid and follow one last circuit with both views in mind. Read the small graph for values, zeros and signs. Read the polar picture for the locations those instructions produce. They are not competing explanations. Each makes a different part of the same equation directly visible.